PHYSICAL REVIEW B VOLUME 27, NUMBER 9 1 MAY 1983

Diffusion-limited aggregation

T. A. WittenExxon Research and Engineering Company, Linden, New Jersey 07036

L. M. SanderPhysics Department, University ofMichigan, Ann Arbor, Michigan 48109

(Received 19 November 1982)

Diffusion-limited aggregation (DLA) is an idealization of the process by which matter ir-

reversibly combines to form dust, soot, dendrites, and other random objects in the casewhere the rate-limiting step is diffusion of matter to the aggregate. We study the processfrom several points of view stressing the fact that it apparently gives rise to scale-invariant

objects whose Hausdorff dimension is independent of short-range details. We show thatDLA has no upper critical dimension. We apply scale invariance to study growth, gelation,and the structure factor of aggregates.

I. INTRODUCTION

The irreversible aggregation of small particles toform clusters is a central problem in many fields ofapplied science; examples are colloids' and coagulat-ed aerosols. The rate-limiting step in aggregation isoften the diffusion of the particles to the surface ofthe aggregate. Similar growth processes occur when

a chemical species precipitates from a supersaturat-ed matrix or when crystals grow from a supercooledmelt. In these cases diffusion of the species towardthe surface (or heat away from it) can be the rate-limiting process. The aggregates formed in all ofthese cases have extremely complicated multi-

branched forms familiar in the case of dust balls, ag-glomerated soot, and dendrites. The complicatedshapes have been associated with the proliferation ofinstabilities induced by the diffusion-limited pro-cess3 (cf. Sec. II).

In a previous publication the authors considereda lattice model which simulates such diffusion-limited aggregation (DLA). This paper contains afurther exploration of the consequences of the

model. The rules of the model are quite simple: Westart with a seed particle at the origin of a lattice.Another particle is allowed to walk at random (i.e.,diffuse) from far away until it arrives at one of thelattice sites adjacent to the occupied site. Then it is

stopped; another particle is launched and haltedwhen adjacent to the two occupied sites, and soforth. An indefinitely large cluster may be formedin this way. A typical structure produced on a two-dimensional lattice is shown in Fig. 1. The modeldoes, thus, produce complicated shapes qualitatively

similar to real dendrites or dust.A new, unexpected result was found ' by direct

measurement of an ensemble of such two-dimensional aggregates: The correlations that arosebetween particle positions seem to be those typicalof a scale-invariant (or dilation-symmetric) object.That is, when viewed with a resolution coarser thanthe size of the aggregating particles the object has nonatural length scale. The density-density correlationfunction for such an object obeys the following:

(p( '+ )p( ') &-

The exponent A is related to the Hausdorff dimen-sion D which characterizes the object by D =d —A.(See Sec. II below. ) In two dimensions, D was foundto be 1.7. Since our original work the results havebeen confirmed and extended to higher dimensionsby Meakin; he also finds evidence for scale invari-ance.

Scale invariance is most familiar in the context ofcritical phenomena where the divergence of thecorrelation length leads to universality (indepen-dence of microscopic length scales and interactions)and correlation functions which have the form ofEq. (1). This symmetry is understood in terms ofthe "renormalizability" of the Hamiltonian govern-ing the statistical mechanics. In this context renor-malization is the process of multiplying certainfields in the theory by the scale factors necessary torender the correlation functions finite when micro-scopic length scales go to zero. The scale factorscontain the information about the Hausdorff dimen-sion of the object in question. A powerful techniquein such studies is the investigation of the dependence

27 5686 1983 The American Physical Society

27 DIFFUSION-LIMITED AGGREGATION 5687

(aj20 Lattice Constants

P',

sl'

krt'it +', +p

L. , -wX,QO[g'i

e &~ f l "le@

~~g gal"~

)It 't'

~l oiooiFeet

(b)

"I. ei goo~Oi

~ /g, t ~ ii Ie

llew'

,P"

L

'l-q~.~i " Fi P', F

~ee C

i i oooo

~i JOI Oei

~~

~oia

0eo !Al'o"

~ ~ ~~ol t40ie

~Oe Oeg

'ieine ~

~oeoeeo

! ~ i OOO ~O ie ~

(c)~0

~OtoottF0' ~~Ottt to

~ ~~ 00~ ~

~ ~

~0 ~000~tte ~' toot 00

F0' OtF00

~ ~~0 ~

~0~000

~Ot ~

, ~0 ~ ~ ~~0000 '' ~ ~ 0' ~ F 0'~ ~

; ~ Otooeet~ ~~ ~~ ~

~ ~~ 0~

, 000 eeate

~0

"Vt 0 ~ &

~ ~Ot~000 ~~0 00

~0 ~ F00'~ 0 ~~0 0000 ~

~ 0~~0

~Ote~ 0~0 ~0

, ~ ~,000 ~0~'00..~..t..... .BOO. . : .

~0,

~000~ ~0~toe

~ ~0~ ~0 ~ ~

,~000 ~ ~0 ~~ ~ ~~0 ~ ~~ ~ 00000000~~to ~ 00~ 0

~ F000F00

FIG. 1. (a) Aggregate of 3000 particles on a square latoo

tice. (b) Screening effect: The first 1500 particles to at-tach to the aggregate are open circles, the rest are dots.(c) Fragments of the same aggregate chosen at random.

are often used.The concepts of upper critical dimension and re-

normalizability apply to the best-known randomdensity profiles: self-avoiding random walks andrandom lattice animals. ' These are qualitativelysimilar to DLA clusters and percolation clusters. "However, there is a fundamental difference, whichwe discuss in Sec. III. The random walks and latticeanimals form an equilibrium ensemble; DLA is de-fined kinetically.

Kinetic models have been much less thoroughlystudied than equilibrium ones. A model similar toours (that of Eden, ' see Sec. II below) has been pro-posed in the context of biology. However, the Edenclusters appear' to lack the anomalous scale invari-ance of DLA. Other kinetic processes have beenproposed which are siinilar to DLA (Ref. 14)though the spatial structure of the objects producedhas not been the focus of the studies.

In this paper we study the DLA model and its di-lation symmetry and illustrate certain aspects whichmake it possibly important both theoretically andpractically. We have been guided in our research bythe successes of the equilibrium theories mentionedabove. The paper is organized as follows: In thenext section we review the origin of the randomnessof the aggregates as it arises from the proliferationof instabilities of the Mullins-Sekerka type. Thenumerical evidence for scale invariance for d=2 ag-gregates is discussed; some elements of universalityare demonstrated by relaxing the constraint that theaggregate absorb perfectly.

In Sec. III we give equations for the time develop-ment of the density p(r). We find that DLA has noupper critical dimension in the ordinary sense. (Thesame line of reasoning shows that Eden clustershave D=d for any d.) These findings show thatDLA is qualitatively different from known criticalphenomena. We have been unable to show whyDLA is dilation symmetric. To show this will re-quire a generalization of renormalization theory yetto be developed.

In Sec. IV we turn to a discussion of some conse-quences of the dilation symmetry. We study thegrowth rate of aggregates, the gelation of a collecoo

tion of aggregates, the dependence of the scatteringstructure factor on D, and the relation of the densityprofile, its projection, and its cross section to D.Section V contains a summary of our conclusions.

II. SCALE INVARIANCE IN DLA

of D on spatial dimensionality d; in commonly stud-ied systems D is simply derivable by dimensionalanalysis above a certain upper critical dimension d, .For dimensions lower than d, expansions about d,

A. Stability analysis

Diffusion is essential in producing the dilationsymmetry observed in DLA. To see this, we consid-

T. A. %'ITTEN AND L M. SANDER 27

er the analogous growth model without diffusion:the "growing animals" of Eden. ' Here every per-imeter site is equally likely to grow at each step.Eden clusters appear to fill a finite, nonzero fractionof space, i.e., D =d (cf. Sec. III); they lack the mul-tibranched structure of DLA. In DLA the exposedends of the clusters grow more rapidly than the inte-rior, because the random walkers are captured be-foge they reach the interior.

To show in detail the origin of the complex struc-tures such as those in Fig. 1, we first point out thatin a certain limit an electrostatic analogy can beused. Consider a random walk which eventuallyadds to the aggregate. I.et u( x,k) be the probabilitythat the walk reaches site x at the kth step. As inany random walk u obeys the following:

u (x,k+1)=—g u(x+ l,k),C

1

(2a)

Bu

Bt=qV u, (2b)

where 1 runs over the c neighbors of x. This, ofcourse, is a discrete version of the continuum dif-fusion equation

on a conductor; thus sharp points grown unstably.In fact, our aggregates look qualitatively like struc-tures (e.g., lightning) which occur due to electricalbreakdown. '5

A graphical illustration of this idea may be seen

by considering the growth of a tower of spheres (seeFig. 2). If a small amount of material is added uni-

formly (as in the Eden model), then the conical en-

velope of the tower is unaffected (except near thetip). However, in diffusive aggregation the growthrate of each sphere is proportional to 1/'R; thus thetower becomes steeper since the smaller spheresgrow more rapidly.

Mullins and Sekerka treat this problem using astability analysis for a slightly deformed sphere. %ecan do a similar analysis for d=2; consider a discwhose radius is given by

r =R +5~cos(m8),

where 5 is small. Outside the disc u is a solutionto Laplace's equation:

u =Aln (r)+8+C~cos(m8)/r™ .

Using u (r) =0 and Eq. (3b) gives at once

where q is the diffusion constant. Also, since walksvisiting perimeter sites terminate there we must setu=0 on the right-hand side of (2a) for these sitesand for cluster sites. The probability that a perime-ter site gains a particle at k + 1 is, as above,

dR/dt =3/8,d5 /dr =(m —l)5 A/R2,

(5/5)I(R /R)=m —1 .

(6a)

(6b)

(6c)

U(x, k+1)=—g u{x+ l,k} .1

(3a}

In the limit of a smooth surface with unit normaln the normal growth velocity V„can be expressed asa gradient:

V„=r)n Vu i, .

Equations (2b) and (3b) are, of course, standard instudies of dendritic growth. In DLA the field u farfrom the aggregate is such as to provide a steady to-tal flux to the surface; this is another boundary con-dition on u.

Now consider the time dependence of u: %'ith asteady flux from far away the only source of timedependence is in the boundary condition {3b) via thegrowth of the aggregate. In our simulations thegrowth is sufficiently slow that it is adequate toneglect Bu /Bt in Eq. {2b), as we discuss in Sec. III.

We are faced, then, with solving Laplace's equa-tion for u in the presence of a boundary where u=0(effectively, a "conducting" surface}. The growthrate V„of any region on the surface is proportionalto the "electric field" (i.e., Vu) there. Now it is wellknown that electric fields are large near sharp points

That is, all deformations for m&1 grow unstablyand, from Eq. (6c), those for rn ~ 2 grow faster thanthe radius itself. Thus tiny amounts of noise willperturb a smooth surface irretrievably. A similartreatment works for three dimensions and higher.This wrinkling instability shows why DLA producesirregular, noncompact shapes.

FIG. 2. Growth of a tower of spheres according to (a)the Eden model and (1) DI A.

27 DIFFUSION-LIMITED AGGREGATION 5689

B. Simulations of the model

Numerical simulations of the DLA process werefirst carried out in two dimensions and reported inRef. 4. A plot of a typical aggregate is in Fig. 1(a).A few more details of the technique may be usefulto mention here. At each stage of the computationit is necessary to launch particles at random, letthem walk randomly, stop them when they arrive ata perimeter site, or to start them over when theywander away. For launching it is sufficient to startwalkers on a small circle circumscribing the existingcluster. To catch errant particles we chose anothercircle twice the radius of the cluster.

The motivation for the location of the circles is asfollows: We are trying to simulate the situation ofisotropic flux very far from the aggregate. We ex-

pect that the field u will be distorted from isotropyon scales of the order of the size of the aggregate.The inner circle can be close to the object becausethe first passage of particles from far away towardthe object is isotropic'; thereafter, to represent thesituation without bias we must not stop particles un-

til they are far away compared to the radius. Wehave performed simulations with the outer circle at3)& the aggregate size. The results were the same

except that they took more computer time.A number of interesting features become evident

in the course of examining the simulations. Theseare shown in Figs. 1(b) and 1(c). In Fig. 1(b) we dis-

tinguish the first half and second half of the parti-cles added. The significant feature is that particlesdo not penetrate to the interior as they are added,but stick to the protruding "arms" even though theinterior appears readily accessible to the exterior.The cluster effectively screens the flux of diffusingparticles.

In Fig. 1(c) we show several small sections of the

large figure positioned at random to illustrate thatthe clusters appear locally homogeneous and isotro-

pic. The appearance of these pictures leads us to thehypothesis that the density correlations within abounded region attain a well-defined limiting formfor indefinitely large clusters, and that this form isinvariant under translations and rotations. In thisrespect the density would be similar to the orderparameter or energy field at a critical phase transi-

tion.To test for scale invariance we use the techniques

originally used by Forrest and Witten to study thescale invariance of metallic smoke aggregates whichseem to be a physical realization of DLA. The cri-terion for scale invariance that we use is that in ascale-invariant object all correlation functions areunchanged up to a constant under rescaling oflengths by an arbitrary factor b:

(p(br, )p(br2) . .p(brk))

=b "(p(r] )p(r2 ) . p(rk ) ) . (7)

In practice, we calculated only the two-point func-tion (p(r&)p(r2)}, whose exponent we call A forshort. Equation (7) leads at once to Eq. (1).

As a cross check we also measured the radius ofgyration R (N) as a function of the number of parti-cles N. It is clear from Eq. (1) that

RN- f (p(0)p(r))d r-R "-R . (8)

Thus the Hausdorff dimension is d —A.To measure the density-density correlations

within the aggregates, we computed the c(r) defined

by

c (r}=—g p(r+ r ')p{ r '} .

The average of c (r) over many clusters positioned atrandom is the correlation function of Eq. (1). Wecomputed c(r) is such a way as to maximize the in-

formation sampled from each image. We Fouriertransformed the density profile of the cluster using afast-Fourier-transform technique, found the powerspectrum, inverted the transform, and averaged overdirections and over narrow ranges (up to about 15%spread) of r. Often we used a matrix of densitieswhich was not identical to the original cluster lat-tice: We overlaid the lattice with a coarser gridoriented at random and summed the cluster pointswithin each cell of the coarse grid to form p(r)This allowed us to treat larger clusters with a givensized matrix at the cost of some short-distance in-formation. This method was calibrated by applyingit to a Koch curve with D constructed to beln3/ln2. The D value obtained from the measuredcorrelation function was correct to within 0.01.

The average c (r}of six aggregates on a square lat-tice is shown in Fig. 3. We note that the measuredc (r) falls below the power-law line when r becomescomparable to the size of the aggregates; note the ar-row which marks the average radius of gyration.The values of D deduced for two-dimensional aggre-gates are displayed in Table I.

Our estimates of D are in good agreement withthose of Meakin who used larger aggregates. The Dvalues are significantly different from those charac-teristic of equilibrium profiles, also displayed inTable I.

C. Universality

One of the most striking features of the power-law behavior observed in critical phenomena is its

5690 T. A. WITTEN AND L. M. SANDER 27

p.4

03-

~p 4~ ~-o.s4

jL

0.15—

0.10-

2.0I I I I

5.O 1p.o 2O.O 5O.O

r {LATTICE CONSTANTS}

FIG. 3. Density correlation function averaged over six

aggregates as a function of distance measured in latticeconstants. The arrow marks the average radius of gyra-tion. The solid line is a least-squares fit over the ranger =3—27. The error bars represent the spread of values

among the six aggregates. u i, =I'IL, (&0}

universality, i.e., its independence of microscopic de-tails of the Hamiltonian. Such behavior is not unex-pected in a situation where statistical properties leadto the buildup of correlations on scales much largerthan those of the microscopic interactions. Thisbehavior should also occur in our case. In this sec-tion we will present evidence that D is universal forDLA in several respects.

First, the long-range properties of our aggregatesshould be independent of the lattice on which we didthe simulation. To check this we produced clusterson a triangular as well as a square lattice. An exam-ple is given in Fig. 4; the Hausdorff dimension forthe aggregates on the triangular lattice, given inTable I, is not significantly different from those on a,square lattice. Meakin did very interesting simula-tions with no lattice at all; and again found the sameD.

We were motivated to investigate a second aspectof universality by comparison with treatments ofdendritic growth. ' Our Eqs. (2b} and (3b} are thesame as in those treatments, but our boundary con-dition u ~, =0 is not, except in the case of vanishingsurface tension. The radius of a real dendrite tip isdetermined by the capillary length I via the Gibbs-Thomson boundary condition

TABLE I. Values of correlation exponent A and Hausdorff dimension D for particles intwo dimensions. Errors are statistical.

DLA: correlation functions of aggregates ona square lattice; average of six clustersof 2079—3609 particles

DLA: correlation functions on a triangularlattice; average of three clusters of1500—2997 particles

DLA: radius of gyration of six clusters of999—3000 particles

Self-avoiding walk in two dimensionsfrom correlations in step density'

Random animals, radius of gyration

Percolation'

'D. S. McKenzie, Phys. Rep. 27C, 37 (1976).bReference 13.'D. Stauffer, Phys. Rep. 54, 1 (1979).

0.343+0.004

0.327 +0.01

0.299

0.667

1.66

1.67

1.70+0.02

1.33

1.54

1.89

27 DIFFUSION-LIMITED AGGREGATION 5691

~0F'

~ ~

FIG. 4. Aggregate of 2500 particles on a triangular lat-tice.

~Isl 4$$OI$t

~0 ee $$Ieaoo~N

~ I~ IO

$ ~et' ~

~ ~'$$$

$~0$ $

, .I ".:":::.."'. .:.. QI

~$~ 0$ $ u

~I ee$$ i' ~$ ~"$1$$$o$

~ OI

1$$'~ I ~,1$

!.1$1)-$:--'1:

~"1$' '$

~M4e$

$1)$$ $$ o

40$$1» ~ ~o$'1 '

ni ~

~ini

T"'1 $e.,J 1$$$$1

I.. I~

I'ii':i!:-):$ lee

~0~0

~u$

where K is the local curvature of the surface. Ourperfectly absorbing clusters correspond to zero sur-face tension, and hence to a diffusing field which isconstant over the surface. We first relaxed the con-dition of perfect absorption by allowing the random-ly walking particle to stick to the aggregate onlyduring a fraction s of its encounters; previously,s=1. This partial sticking introduces a lengthparameter into the diffusion equation.

To see this we work in one dimension, and write

u(l, k +1)=(1—s)u (O,k)+ $ u (2l,k), (1la)

u(O, k+1)=—,u(l, k) . (1 lb)

Here 0 labels the perimeter site and 1,2l, the freespace. In steady state we may drop the k label andmanipulate Eqs. (1 la) and (1 lb} to find

su (l)=u (2l) —u (I),or, in a continuum limit, as above,

su ~, =ln Vu ~, .

(12a)

(12b)

The fixed logarithmic derivative of u at the surfaceintroduces a new length scale,

A, =I/s, (13}

which we may adjust at will. This new length playsthe role of the capillary length in conventional treat-ments. Indeed, we can use our electrostatic analogyto reinterpret Eq. (12b). Consider a spherical sur-face; then if A, &&R, Vu ~, -u /R and to first or-der in A, we may put

u i, =u„A/R, (14)

i.e., a condition of the form of Eq. (10).To gain some appreciation of what Eq. (12b) im-

plies we repeated the stability analysis with the new

FIG. 5. Aggregate of 3000 particles with s=0.5. Notethe thickening of the "branches. "

boundary condition. We find, for d=2 [compareEq. (6)],

d5~/dt =(m —1)5 A/R(R+mA),

(5/5)/(R /R) =(m —1)R/(R +mA, ) .

(15a)

(15b)

Note that for A, &&R/m, d5/dt is suppressed com-pared to its value in Eq. (6). It does not change signas in the usual dendritic growth case, but if R & A,,for any m the distortions will become smaller rela-tive to R as time proceeds instead of larger. Sincethe instability leading to DLA is quenched forR & A, , we expect the DLA density profiles to be uni-form over distances R &A,. For larger R, the insta-bilities are again present, and we expect the densityprofiles on this scale to be multibranched, like thoseof the simple DLA. The same conclusion holds inthree dimensions.

Using these insights we created a number of ag-gregates with s&1. Plots of a cluster of this typeare shown in Fig. 5, and its correlation functions aregiven in Fig. 6. The correlation function does havethe form we hoped for: At sinall distances c(r) isroughly constant and represents the thickening ofthe "branches" of the object; at large distances (butstill smaller than the total size of the object) thepower law is unchanged. The change in behaviorfrom constant to power law occurs at a distancewhich increases as s decreases, in accord with ourprediction of crossover for R =A, -1/s.

We also did simulations with a more ad hoc typeof boundary condition. We take a sticking probabil-

5692 T. A. WJ J JEN AND L. M. SANDER 27

1.0-

0.5-0.40.3

0.2~J

liL'

0.1 ~Melgpg

C(r)

S = 0.5

FIG. 7. Aggregate of 3000 particles with t=0.5.

I

2.0I I I I

5.0 10.0 20.0 50.0r (LATTICE CONSTANTS)

5.FIG. 6. Correlation function for the aggregate of Fig.

ity which depends on the number of bonds to filledneighbor sites B; for a square lattice we put

large-scale properties of large aggregates.We may describe any given cluster i on a lattice

by a density field [p(x)] which is 1 at an occupiedsite and zero elsewhere. Aggregation processes gen-erate a large class of clusters i, each with a relativeprobability f~. Thus, e.g., in the ensemble of "ran-dom animals, "

any connected pattern of 1's hasfi= 1; any disconnected pattern has fr=0. In theDLA model the f; for a given cluster cannot be ex-pressed directly in terms of the p field. One must

s=t (16)

where t is an adjustable parameter less than 1. Thusa hole (with B=3) is far more likely to fill than apoint (with B=1).

A simulation for t=0.5 is shown in Fig. 7, andcorrelation functions for several values of t in Fig. 8.Once more, small t introduces a minimum lengthscale for the object, but the long-range propertiesseem to be unaltered.

The simulations of Meakin in higher dimensionsseem to bear out our hypothesis of scale invariance.For d =3 he finds D=2.51+0.06 by examining onlythe radius of gyration. We have presented' data forc (r) in d=3, which is in agreement with Ref. 7.

1.0-

0.50.40.3

0.2

0.1

C(r)

~-0.340

III. GENERAL FORMULATION

The computer simulations described above, if re-peated for all possible random walks, would generatea statistical ensemble of clusters. In this section weformulate equations for the time development ofthis ensemble. We argue that these equations maybe simplified considerably without affecting the

I

2.0I I I I

5.0 10.0 20.0 50.0r (LATTICE CONSTANTS)

FIG. 8. Correlation functions for three aggregates witht(1.

27 DIFFUSION-LIMITED AGGREGATION 5693

also know the f; for other clusters; it does not be-

long to an equilibrium ensemble like that of the ran-dom animals. The f~ are determined instead by atime-development equation.

We consider the time derivative df;pat for a par-ticular cluster i. In a given instance the cluster i isproduced by the growth of some other cluster, i —x,equal to i but for the absence of the site at x. Thegrowth of i from i —x occurs with an average ratewhich we denote v; „(x). The (positive) contribu-tion to f; from these growth processes is

QU; „(x)f, „.xCi

(17)

The cluster i also disappears from the ensemble at acertain rate, as larger clusters grow from it. Theclusters thus produced have one particle added atsome site x, and we denote them as i+x. Combin-ing the processes which increase and which reduce

f~, we have

= g v; „(x)f; „—g v;(x) f;, (18)x6i xCBL

where we have used Bi to denote the sites on the per-imeter of i. By contrast, a thermal equilibrium en-semble with temperature P ' and energy E; has

df; IdIB=E;f;; there is no coupling of different f, .Other growth models are governed by time-

development equations of the form of Eq. (18). Forthe Eden model, ' the v factor is taken independentof i and x, so that all perimeter sites grow at thesame rate.

The distinctive feature of diffusive growth is thatthe growth rate v is the flux of a diffusing field u, asdiscussed in Sec. II above: the effect of the absorb-

ing cluster on the diffusing field u may be expressed

by a boundary condition at the cluster surface. In-stead, we allow the field u to occupy any site, but in-troduce a strong absorption at perimeter sites. Sucha site x is indicated by the function

P(x)=[1—p(x)] 1 —/[1 —p(x+1)]

atu

(20)

Evidently, in the limit of large Z the perimeter sitesbecome an absorbing boundary, where u~0. Forany Z the local rate of absorption Zu (x) is also theaverage rate v( x ) of cluster growth there.

The equations for u and f; determine the proba-

(19)

which is 1 for perimeter sites and 0 elsewhere. Thusthe diffusion equation for u may be written

bility of a given cluster as a function of time givensuitable boundary conditions. The equation for f;requires an initial condition, e.g. , f; =0 at t=0, ex-cept for f~

——1 for the single particle at the origin.The equation for u requires a boundary conditionsuch as u=1 for R =R where R is a large radius.

The equation for u may be simplified by recallingthat Bu/Bt does not play a crucial role. The field u

changes most rapidly with time near a point x wherea site has just grown. The effect of the new ab-sorbers is appreciable over distances r of the order ofthe lattice spacing —an arbitrarily small fraction ofthe growing sites. Thus the u field relaxes every-where to its new steady-state value in a time in-

dependent of the cluster size. Within this time, onlya small amount of new growth occurs and only anegligible part of this lies in the region around xwhere the time dependence is appreciable. ' Whenthe growth rate per surface site is negligible com-pared to the diffusive relaxation time, the timedependence of the relaxation may be neglected. Thisamounts to neglecting the Bu/Bt term. Without thisterm, the u field relaxes instantly to the clusterwhich exists at that time. We shall adopt this "adia-batic" approximation in the sequel so that v and u

depend on the cluster, but not otherwise on time.We may also formulate DLA to derive an equa-

tion for the random variable p(x) itself. The densityp(x) grows in discrete units at an average rate pro-portional to v(x). We let q(z) be a random noisevariable equal to 1 with probability z and zero withprobability 1 —z. Then in an elementary time stepht,

p(x, t+ht)=p(x, t)+q(v ht)P(x),

so that

q(v (x)b t )

ht ht

(21)

(22)

where the resolution function w sums to unity andhas a spatial extent of order a. This smoothing willnot alter correlations at distances much larger thana, yet it gives a smooth density which is easier totreat. Further, any field p with inverse power corre-lations will decrease upon averaging. We expectthen, that sufficiently high powers of the field maybe neglected after averaging, as in field theories ofphase transitions. We are thus led to expand out theexpression (19) for P:

Our interest here is to understand the correlationsof the p field on length scales much larger than thelattice spacing. Accordingly, we may use a continu-um description for a smoothed field

p, (x)—= g w(x —y)p(y),

5694 T. A. WrrTEN AND L. M. SANDER 27

C

P(x)=[1—p, (x)] 1 —g [1—p, (x+1}] =[1—p, (x}] gp, +0{p,}n=1

p. p—Xp. +0 (p )

=c V p, +cp, +c(c —3)p, /2+0(p, V p, )+0(p,') . (23)

dpi'V Q~=

dt

(24)

where r, and g, are renormalized parameters de-

pending on the averaging scale. We may anticipateseveral qualitative features of these equations. Wefirst investigate the transparency of the density pro-file to the diffusing field u. To this end, we average

p, on a scale comparable to the cluster radius R, andtreat it as a simple absorber of the field u:

V u =Zup, (r) . (25)

This extreme averaging would be justifiable as longas the cluster is transparent to the diffusing field.Within the radius R, p, is roughly constant and ufalls off exponentially in a characteristic skin depthgiven by 1/h =Zp, (r) In a large. cluster, p, is ex-pected to become arbitrarily small, so that the skindepth goes to infinity. But if the cluster is to betransparent, we require that h/R —+00, so thatR p~{r) must go to zero. For clusters with a Haus-dorff dimension D, pa{r)-N/R -R . Thus(R/h) -R' + ' . This result is natural since thepath of our random walker has Hausdorff dimen-sion 2. The R' + counts the number of intersec-tions' of the cluster and the random walker, withinradius R, assuming these are independent. For2+ D & d, the number does not diverge with R, thenumber of contacts for potential absorption remainsfinite, and the cluster remains transparent to thediffusing field. But in the reverse case we expect thecluster to be opaque, with new growth occurring

In addition the averaging runs over many values ofthe random variables q, so it seems plausible to re-place this variable by its average v. To perform thisaveraging systematically would require a full-fledged renormalization treatment. We defer such adetailed treatment for a planned future paper. Ourpresent aim is to formulate an equation for the aver-

aged fields which incorporates the above expecta-tions. Replacing n by its average v, and using theexpanded form of the P(x) term, we postulate that

dpi' 2

dt=Zua(V Pa+raPa+gaPa+ ' ' ' } t

only on the periphery. This opacity is observed tobe the case in our simulations where indeed

2+D &d.Since growth occurs only within a thin layer, we

may average the fields only over distances smallerthan this layer. In the growing layer the gradient ofp is large and it is not clear that the V p can beneglected in the growth equation. No correct sim-plification of this equation is apparent to us.

To shed light on our growth process, we havestudied its behavior in high-dimensional space withthe aim of deriving its asymptotic D. The motiva-tion for this approach is the simple behavior ofknown aggregates and other dilation-symmetric pro-files above a certain upper critical dimension. Thecase of random animals illustrates this point. Ran-dom animals or randomly branched chains may bereadily treated under the assumption that there isonly a single path on the object connecting any pairof points; such objects have D=4. One may investi-gate the validity of the assumption by estimating thenumber of contacts between two parts of the objectwhich meet at the origin, assuming the two are in-dependent. The density of points in each of the twoparts falls off as r; thus the density of crossingsgoes as r ' "' The nu.mber N(R) of crossingswithin radius R goes as

R 2(D —d)+d R 2D —d

The number grows indefinitely with R if 2D &d.Thus the argument giving D=4 requires 2D &d, ord& 8. This criterion gives an upper critical dimen-sion of 8. A more standard field theoretic argu-ment ' gives the same result. Random linear chainshave D=2 if intersections of links are allowed. Bythe same reasoning these intersections are negligiblewhen d &2D=4. Thus 4 is the upper critical di-mension in agreement with polymer field theory.

For the DLA problem, there is no upper criticaldimension in this sense. We show this by assumingthe reverse and arriving at a contradiction. Supposethat D attains an asymptotic value for large d. Since2D & d the cluster may be treated as a treelike struc-ture, where self-crossings are not appreciable. Since2+ D & d, the aggregate is, in addition, transparent

27 DIFFUSION-LIMITED AGGREGATION 5695

to the diffusing field from which it grows, by the ar-gument above. We may thus simplify the growthmodel considerably. Since the field u is essentiallyuniform within the cluster, we allow all sites adja-cent to cluster sites to grow with the same probabili-ty. Since cluster intersections are assumed negligi-ble, we may allow multiple occupancy of any latticesite. These simplifications give a model which canbe explicitly solved. We will see that the radiusgrows only logarithmically with particle number,violating the hypothesis.

To show this, we consider the ensemble of n-siteclusters grown from the origin. The n clusters aregrown by adding each successive particle indif-ferently to the (n —1)c sites adjacent to the existingones. The number G (n) of n-site clusters is thus

D

6-

OO

~....DIFFUSION-LIMITEDAGGREGATION Qr

0,/

//'

/

I I I

4 6

I I I I n///

Qr0/

///

//

//// /

NIMALS

G(x,n)= QS(x+1,n —1), (26)

where S(y, m) is the number of m-particle clusterswith an arbitrary particle at y. The particle at ymust be the ith particle for some i (m. There areG ( y, i) ways to arrange the first i particles, and

ic(i+1)c (m —1)c =G(m)/G(i)

ways to arrange the rest. Thus

S(y,m)= g G(y, i) .G(m)

i=i G i)

Inserting this into Eq. (26) for G (x,n), we find

(27)

G(x, n) G(n —1}+ +' G(x+ l, i)G(n) G(n} -. . . G(i)

The left-hand side is the probability that the nth siteis at x; we denote it by g ( x,n); thus

Jl

g(x, n+1)=—g g g(x+ l, i) .cn

(29)

This may readily be transformed into a differentialequation by multiplying by n and taking an n differ-ence:

n (x,n)=V2g .BgBn

In this diffusion equation, inn plays the role of time.Thus, e.g., the second moment R2 of the distributionincreases as inn.

The radius grows more slowly with n than for anyfinite D: The D of this model is 00, contradicting

(n —1}cG(n —1}=c" '(n —I}!.

The number G(x, n}, with the nth site at x is given

by

FIG. 9. Hausdorff dimension D of the density in vari-

ous random aggregates as a function of spatial dimension

d. The values for DLA are from Ref. 4 (d=2) and Ref. 7(d=3). The two solid lines indicate the approximatebehavior of D(d) for random animals and self-avoiding

walks. The boundaries of the finite density domain

(D &d), the transparent domain (D &d —2), and thenon-self-intersection domain (D &d/2) are indicated bydashed lines.

the hypothesis that it attains a finite asymptoticvalue. We conclude that there is no asymptotic D indiffusion-limited aggregation (or in the Edenmodel). The same argument shows that D & d l2 forlarge d.

Given this, there are two possibilities for theasymptotic behavior of D (see Fig. 9). One is thatD &d —2, so that the clusters are transparent. Inthis case the high-d behavior is that of the Edenmodel. Assuming this, we are led again to a con-tradiction.

Our supposedly transparent clusters have D &d,and hence the perimeter of the cluster has the sameD as the cluster itself. The number of perimetersites is proportional to the number of cluster sites,with some proportionality factor less than the coor-dination number c. Thus we may repeat the deriva-tion of G(n), S(y,m), and G(x,n) above for theEden model, replacing c by an effective coordinationnumber c. The sum over neighbors in the equationfor G(x,n) must be multiplied by c/c, since onlythis fraction of sites adjacent to a cluster site con-sists of perimeter sites. These modifications do notchange the qualitative behavior of the growth ofthese clusters as compared to the multiple occupan-cy clusters of our original model. We thus arrive

5696 T. A. WITTEN AND L. M. SANDER 27

again at the contradictory result that D~oo, indi-cating that the original hypothesis of D &d wasfalse. This shows that diffusion-limited aggregatescannot be transparent to the diffusing field, so thatd —2(D &d. It also shows that Eden-model aggre-gates must be compact, as has been previouslyshown numerically' ' for d =2 and 3.

It appears from this study that our Monte Carlosimulations of DLA correspond to a well-definedstochastic model which is amenable to treatment ina continuum language. It appears further that themodel has no upper critical dimension in the usualsense. Instead, d —D may attain an asymptotic lirn-it between 0 and 2, so that the aggregates are alwaysopaque to the diffusing field.

IV. CONSEQUENCES OF SCALEINVARIANCE IN DLA

S(q)= f d r(p(r'+r)p(r'})e'7'' . (32)

The power-law correlations of p in some range of rgive rise to a conjugate power behavior in S(q) inthe range q-r '. Here S(q) is proportional to thenumber of scatterers within a distance q

' of an ar-bitrary scatterer. Thus S(q)-q D in this range, asone may check by direct substitution into the defini-tion of S(q). Since the aggregates are asymptotical-ly opaque to a scattering wave, a weakly scattered

A number of observable properties of objectsgrowing via DLA can be deduced simply from the(assumed) scale invariance of process. We describehere how aggregates should appear in projection andcross section, and how they should scatter x rays orneutrons. We also discuss the gelation expectedwhen many aggregates grow together.

Projected onto a plane, a diffusion-limited aggre-gate with D&2 should appear opaque. The project-ed density

Ro(x,y)= f p(r)dz-R (31)

goes to infinity with the size of the objects, and thelocal density correlations do not survive in the pro-jected density. On the other hand, the cross sectionof a diffusion-limited aggregate evidently has(p(r')p(r'+r)) equal to that of the full aggregate,so that the exponent A is unchanged in the cross sec-tion.

A direct way to observe these correlations is bydiffracting a wave from the aggregates. Thismethod has been successful in detecting power-lawcorrelations in random-coil polymers. The dif-fracted intensity as a function of wave vector q mea-sures the structure function

wave is needed to probe the whole volume of the ag-gregate.

The growth rate of an aggregate in three dimen-sions can also be readily deduced. We consider theregime where the number of free particles is muchgreater than the number of aggregated particles, sothat u(oo) is constant in time. Then at large dis-tances r, the diffusing field u obeys

u (r) =u ( a)o[1 —R (N)/r],

g V u ~, = riu ( ao )R /r

(33)

(34)

The capture "radius" R (N} must be proportional tothe radius of gyration since the aggregates areopaque to the diffusing field; the quantity in Eq.(34) is the flux density incident on the object. Thetotal rate of aggregation is given by

dN/dt =R riu ( ao ) f d r /r

-R (N) -N'~+, (35)

Solving the equation gives N-t ' ". This lawgives the asymptotic growth rate of an aggregate.

In a similar way one may predict gelation (i.e., in-

tersection) of a collection of growing aggregates.Suppose that cp is the concentration of nucleated ag-gregates. They will start to intersect and form a gelwhen cpR 1. The concentration of monomers(the "particles" ) is

c=N(R)/R =R =c' (36)

In three dimensions this gives c-cp . Thus themass of the gel may be made indefinitely small byreducing the concentration of seeds. This effect isaugmented by the percolation effects present in ordi-nary gels.

V. SUMMARY

The DLA model which we have discussed in thispaper seems to be applicable to a wide variety ofphenomena which at first sight are unrelated. Assuch, its general properties are worth examining indetail; it is to this task that we have addressed our-selves.

The most striking features that we have discussedare the long-range, universal properties associatedwith scale invariance. It is worth repeating explicit-ly that these long-range properties do not arise fromlong-range forces: Whatever occurs in real dust,smoke, etc., our simulations show that short-rangeforces can build up long-range correlations, just asin the case of critical phenomena.

The remaining theoretical problems associatedwith our model seem formidable. We believe itwould be very interesting and useful to understand

27 DIFFUSION-LIMITED AGGREGATION 5697

on a deeper level the apparent scale invariance thatwe observe; such understanding has been arrived atin other apparently similar problems (such as that ofthe self-avoiding random walk). We have shownthat DLA is much too different from critical phe-nomena to be treated by a direct translation of themethods used there. The lack of an upper criticaldimension for the theory is one signal of our diffi-culty.

Some promising approaches have appeared andshould be pursued. Rikvold has reduced DLA to asimpler problem by schematically representing the"screening" effect by a reduction of the flux onto apoint according to a formula involving the numberof neighbors in shells surrounding the site. This is afurther simplification of the mean-field treatment ofSec. III which may be useful if it indeed contains allthe relevant physics (which is not clear to us at the

moment). Gould, Family, and Stanley are usingthe methods of real-space renormalization for DLA(and other similar problems). Of course, this ap-proach assumes the existence of scaling symmetry; itshould allow an approximate calculation of D.

ACKNOWLEDGMENTS

We would like to thank L. Auvray, Z. M. Cheng,P. G. de Gennes, H. Gould, R. Richter, H. Muller-Krumbhaar, T. M. Sanders, R. Savit, and L. S.Schaefer for useful discussions. One of us (T.W.) isgrateful to the College de France for hospitality.This work was supported in part by the Centre ¹

tional de la Recherche Scientifique and by the ¹

tional Science Foundation Grants Nos. DMR-78-25012, DMR-82-03698, and DMR-80-12867 (poly-mer program}.

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323 (1963).4T. A. Witten, Jr. and L. M. Sander, Phys. Rev. Lett. 47,

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(1979).B. Mandelbrot, Fractals, Form, Chance, and Dimension

(Freeman, San Francisco, 1977).~P. Meakin, Phys. Rev. A 27, 1495 (1983).D. Amit, Field Theory, the Renormalization Group, and

Critical Phenomena (McGraw-Hill, New York, 1978).P. G. de Gennes, Phys. Lett. 38A, 339 (1972).C. Domb, J. Phys. A 9, L141 (1976); see also Ref. 11and references therein.D. Stauffer, Phys. Rep. 54, 1 (1979).

t~M. Eden, in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, edited

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See Ref. 12 and H. P. Peters, D. Stauffer, H. P. Holters,and K. Loewenich, Z. Phys. B 34, 399 (1979).

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Bois, Macromolecules 13, 939 (1980); H. J. Hermann,D. P. Landau, and D. Stauffer, Phys. Rev. Lett. 49,412 (1982).

'5We are grateful to Loic Auvray for pointing this out.See E. D. Forster, J. Electrostatics 12, 1 (1982).M. E. Sander (private communication).

'7J. S. Langer and H. Muller-Krumbhaar, Acta Mettall.26, 1081 (1979); 26, 1689 (1979); H. Muller-Krumbhaarand J. S. Langer, ibid. 26, 1697 (1978).

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2~T. A. Witten, Jr., in International Union of Pure andApplied Chemistry 28th Macromolecular Symposium,University of Massachusetts, Amherst, Massachusetts,1982, edited by J. Chien, R. Lenz, and O. Vogl(IUPAC, New York, 1982).J. P. Cotton, D. Decker, B. Farnoux, G. Jannik, R.Oker, and C. Picot, Phys. Rev. Lett. 32, 1170 (1974).P. A. Rikvold, Phys. Rev. A 26, 647 (1982).H. Gould (private communication); H. Gould, F. Fami-ly, and H. E. Stanley, Phys. Rev. Lett. 50, 261 (1983).